Does your employer have an office in Georgia, and do you work in that office, or are you telecommuting to New York? If you are telecommuting, is it because you want to live in Georgia, or because your employer wants you to be located in Georgia for business reasons?
Did you notify the payroll department that you moved to Georgia? If they have a New York address for you, that would certainly lead them to think that you live in New York. I don't think most payroll systems are set up to handle a mailing address that is different from the employee's residence.
They certainly should not be withholding New York City tax if you are not a NYC resident. Whether they should be withholding NY State tax depends on your answer to the telecommuting question. If the New York withholding is wrong, you will have to ask the payroll department how you can fix it. It might be a bit late to be correcting withholding that started in January. If they won't correct it, you will have to file a New York tax return and try to convince New York State that you did not live or work there. But it's going to be hard to convince them of that if you have a New York address.
